For the Galois closure
X
gal
X_{\textrm {gal}}
of a generic projection from a surface
X
X
, it is believed that
π
1
(
X
gal
)
\pi _1(X_{\textrm {gal}})
gives rise to new invariants of
X
X
. However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of
π
1
(
X
gal
)
\pi _1(X_{\textrm {gal}})
that depends on
π
1
(
X
)
\pi _1(X)
and data from the generic projection only. In all known examples, this quotient is in fact isomorphic to
π
1
(
X
gal
)
\pi _1(X_{\textrm {gal}})
. As a byproduct, we simplify the computations of Moishezon, Teicher and others.